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Grundzüge Der Mathematischen Logik by Heinrich Scholz, Gisbert Hasenjaeger

Grundzüge Der Mathematischen Logik

by Heinrich Scholz , Gisbert Hasenjaeger
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Book cover type: Paperback
  • ISBN13: 9783642948152
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 504
  • Original Price: EUR 70.08
  • Language: German
  • Edition: Softcover Repri
  • Item Weight: 694 grams
  • BISAC Subject(s): General

1. Prolegomena 1. Die Logik, die in diesem Lehrbuch entwickelt wird, ist bestimmt durch die folgenden Kennzeichen: (1) Sie fuBt auf derselben Ontologie wie die von erkennbaren Wider- sprlichen befreite und in diesem Sinne vertretbare klassische Mathe- matik. Flir diese Ontologie ist charakteristisch die Grundvoraussetzung, daB die Objekte der Mathematik und mit ihnen die mathematischen Bereiche an sich existieren, wie die platonischen Ideen. Mit Bezug auf diesen An-sich-Charakter sprechen wir von einer platonischen Ontologie. Flir diese Ontologie existieren die unendlichen Bereiche beliebig hoher Machtigkeit als fertig vorliegende Objekte in derselben Art wie die durch Aufzahlung ihrer Elemente erfaBbaren endlichen Mengen und in gleichem Range mit ihnen. Die weittragenden Folgen dieser Auffassung sind in zwei Haupt- punkten konzentriert. Erster Hauptpunkt: die Beurteilung der Potenz- mengen. Mit den abzahlbaren Mengen existieren, genauso wie im end- lichen Falle, auch ihre Potenzmengen, mit dem ihnen zukommenden Charakter der Uberabziihlbarkeit. Das Uberabzahlbare steht also gleich- berechtigt neben dem Abzahlbaren. Zweiter Hauptpunkt: der flir den Platonismus charakteristische Gehalt des ausgeschlossenen Dritten. Es genligt hier, die mengentheoretische Formulierung dieses Prinzips ins Auge zu fassen. Sie besagt, daB eine flir die Elemente einer beliebigen Menge, also mit EinschlieBung der unendlichen Mengen von einer be- liebigen Machtigkeit erklarte Eigenschaft allen Elementen der Menge zukommt oder es gibt (wenigstens) ein Mengenelement, dem sie nicht zukommt, unabhangig davon, ob ein solches Element angegeben werden kann oder nicht. Ein Anwendungsfall ist das der elementaren Zahlen- theorie angeh6rige Prinzip der kleinsten Zahl.

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